Optimal. Leaf size=38 \[ \frac{(a+b x)^7 (A b-a B)}{7 b^2}+\frac{B (a+b x)^8}{8 b^2} \]
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Rubi [A] time = 0.0151013, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ \frac{(a+b x)^7 (A b-a B)}{7 b^2}+\frac{B (a+b x)^8}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (A+B x) \, dx\\ &=\int \left (\frac{(A b-a B) (a+b x)^6}{b}+\frac{B (a+b x)^7}{b}\right ) \, dx\\ &=\frac{(A b-a B) (a+b x)^7}{7 b^2}+\frac{B (a+b x)^8}{8 b^2}\\ \end{align*}
Mathematica [B] time = 0.03453, size = 122, normalized size = 3.21 \[ \frac{1}{56} x \left (70 a^4 b^2 x^2 (4 A+3 B x)+56 a^3 b^3 x^3 (5 A+4 B x)+28 a^2 b^4 x^4 (6 A+5 B x)+56 a^5 b x (3 A+2 B x)+28 a^6 (2 A+B x)+8 a b^5 x^5 (7 A+6 B x)+b^6 x^6 (8 A+7 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 145, normalized size = 3.8 \begin{align*}{\frac{B{b}^{6}{x}^{8}}{8}}+{\frac{ \left ( A{b}^{6}+6\,Ba{b}^{5} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,Aa{b}^{5}+15\,B{a}^{2}{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,A{a}^{2}{b}^{4}+20\,B{a}^{3}{b}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,A{a}^{3}{b}^{3}+15\,B{a}^{4}{b}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,A{a}^{4}{b}^{2}+6\,B{a}^{5}b \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,A{a}^{5}b+B{a}^{6} \right ){x}^{2}}{2}}+A{a}^{6}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.977604, size = 192, normalized size = 5.05 \begin{align*} \frac{1}{8} \, B b^{6} x^{8} + A a^{6} x + \frac{1}{7} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{7} + \frac{1}{2} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{6} +{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{5} + \frac{5}{4} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{4} +{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.3136, size = 316, normalized size = 8.32 \begin{align*} \frac{1}{8} x^{8} b^{6} B + \frac{6}{7} x^{7} b^{5} a B + \frac{1}{7} x^{7} b^{6} A + \frac{5}{2} x^{6} b^{4} a^{2} B + x^{6} b^{5} a A + 4 x^{5} b^{3} a^{3} B + 3 x^{5} b^{4} a^{2} A + \frac{15}{4} x^{4} b^{2} a^{4} B + 5 x^{4} b^{3} a^{3} A + 2 x^{3} b a^{5} B + 5 x^{3} b^{2} a^{4} A + \frac{1}{2} x^{2} a^{6} B + 3 x^{2} b a^{5} A + x a^{6} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.092434, size = 148, normalized size = 3.89 \begin{align*} A a^{6} x + \frac{B b^{6} x^{8}}{8} + x^{7} \left (\frac{A b^{6}}{7} + \frac{6 B a b^{5}}{7}\right ) + x^{6} \left (A a b^{5} + \frac{5 B a^{2} b^{4}}{2}\right ) + x^{5} \left (3 A a^{2} b^{4} + 4 B a^{3} b^{3}\right ) + x^{4} \left (5 A a^{3} b^{3} + \frac{15 B a^{4} b^{2}}{4}\right ) + x^{3} \left (5 A a^{4} b^{2} + 2 B a^{5} b\right ) + x^{2} \left (3 A a^{5} b + \frac{B a^{6}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10909, size = 196, normalized size = 5.16 \begin{align*} \frac{1}{8} \, B b^{6} x^{8} + \frac{6}{7} \, B a b^{5} x^{7} + \frac{1}{7} \, A b^{6} x^{7} + \frac{5}{2} \, B a^{2} b^{4} x^{6} + A a b^{5} x^{6} + 4 \, B a^{3} b^{3} x^{5} + 3 \, A a^{2} b^{4} x^{5} + \frac{15}{4} \, B a^{4} b^{2} x^{4} + 5 \, A a^{3} b^{3} x^{4} + 2 \, B a^{5} b x^{3} + 5 \, A a^{4} b^{2} x^{3} + \frac{1}{2} \, B a^{6} x^{2} + 3 \, A a^{5} b x^{2} + A a^{6} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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